SU326 P30-11

SOME MODIFIED~IGENVALUE PROBLEMS

BY

GENE H. GOLUB

STAN-CS-234-71AUGUST 1971

COMPUTER SCIENCE DEPARTMENT

School of Humanities and Sciences

STANFORD UNIVERSITY

su326 PRO-u

SOME MODIFIED EIGEWALUE PROBLEMS*

Gene H. Golub=

*Presented by invitation at the Dundee Conference on NumericalAnalysis at Dundee, Scotland, in March, 1971.

**Computer Science Department, Stanford University, Stanford, California94305 l This work was supported in part by grants from the NationalScience Foundation and the Atomic Energy Commission.

tli!ABm OF CONTENTS

0 .

1.

2.

39

4.

5-

6.

70

Introduction and notation

Stationary values of a quadratic form subject to linear constraints-.

Stationary values of a bilinear form subject to linear constraints

Some inverse eigenvalue problems

Intersection of spaces

Eigenvalues of a matrix modified by a matrix of rank one

Least squares problems

Gauss-type quadrature rules with preassigned nodes

Abstract

We consider the numerical calculation of several eigenvalue problems

which require some manipulation before the standard algorithms may be-.

used. This includes finding the stationary values of a quadratic form

subject to linear constraints and determining the eigenvalues of a matrix

which is modified by a matrix of rank one. We also consider several

inverse eigenvalue problems. This includes the problem of computing the

Gauss-Radau and Gauss-I,obatto quadrature rules. In addition, we study

several eigenvalue problems which arise in least squares.

0. Introduction and notation

In the last several years, there has been a great development in

devising and analyzing algorithms for computing eigensystems of matrix

equations. In particular, the works of H. Rutishauser and J. H. Wilkinson

have had great influence on the development of this subject. It often

happens in applied situations that one wishes to compute the eigensystem

of a slightly modified system or one wishes to specify some of the eigenvalues

and then compute an associated matrix. In this paper we shall consider some

of these problems and also some statistical problems which lead to interesting

eigenvalue problems. In general, we show how to reduce the modified problems

to standard eigenvalue problems so that the standard algorithms may be used.

We assume thatthe reader has some familiarity with some of the standard

techniques for computing eigensystems.

We ordinarily indicate matrices by capital letters such as A , B , A ;

vectors by lower case letters such as x , y , a,, and scalars by lower caseM m

letters. We indicate the eigenvalues of a matrix as h(X) where X may be

an expression, e.g., h(A2+I) indicates the eigenvalues of A2+I , and in

a similar fashion we indicate the singular values of a matrix by o(X) .

Usually we order the eigenvalues and singular values of a matrix so that

Al(A) 5 h2(A) 5 . . . 5 $(A) and al(A) ,< 02(A) 5 . . . oN(A) . We assume

that the reader has some familiarity with singular values (cf. [g])*

1. Stationary values of a quadratic form subject to linear constraints

Let A be a real symmetric matrix of order n , and c a givenCITvector with c c = 1 .WN

In many applications (cf. [lo] it is desirable to find the

stationary values of . .

TXAXN N

subject to the constraints

Txx=1NN

(1.1)

(14

(1.3)

Let

cp(x) = xTAx - lLxTx + T2px c W)N m - cIc1 GIN

where ob 4 are Lagrange multipliers. Differentiating (1.4), we are

led to the equation

Ax -⌧⌧+pc=o l (le5)LI H c1 c1

Multiplying (1.5) on the left by cT and using the condition thatH

II IIE 2 = 1 , we have

TP= -cAx . (14

N N

Then substituting (1.6) into (1.5), we obtain

PAX = ?LXM m (1*7)

where P = I -ccT . Although P and A are symmetric, PA is notNW

necessarily SO.

Note that2

P = P , so that P is a projection matrix. Thus

h(PA) = A(P2A) = A(PAp) l

The matrix PAP is symmetric and hence one can use one of the standard

algorithms for finding its eigenvalues. Then if

K = PAP

and if

KZi = hiZi tN

it follows that

X. = Pfi ( i-1= 1,2,...,n) .

At least one-eigenvalue of K will be equal to zero, and c will beN

an eigenvector associated with a zero eigenvalue.

Now suppose we replace the constraint (1.3) by the set of constraints

CTx =o (l-8)w

where C isan nxp matrixofrank r. It can be verified that if

f = I - cc- &9>

where C- is a generalized inverse which satisfies

cc-c = c(1.10)

cc- = (CC-)T

then the stationary values are eigenvalues of K = PAP . At least r

of the eigenvalues of K will be equal to zero, and hence it would be

desirable to deflate the matrix K so that these eigenvalues are

eliminated.

l-2

By permuting the columns of C , we may compute the orthogonal

decomposition

C = QT ; ; TT[ 1 (1J-l>where R is an upper triangular matrix of order r , S is rx(p-r) ,

QTQ = In > and TT is a permutation matrix. The matrix Q may be

constructed as the product of r Householder transformations (cf. [8]).

A simple calculation shows

P Q

--_

and thus

WA@ = h(QTJQ,AQTJQ)

= A(JQAQTJ) l

Then if

(1.12)

where Gilis an rxr matrix and G22 is an (n-r)x(n-r) matrix,

JQAQTJ =0 0[ 1O G22 .

Hence the stationary values are simply the eigenvalues of the

(n-r) x (n-r) matrix G22 . Finally if

G22 zi II= hiZi (i = 1,2,...,n-r) ,

l-3

then

X.-1

= QT

c I

;n-r 5 l

The details of the algorithm are given in [lo].. .

From equation (1.13) we see that h(G) = h(A). Then by the Courant-

Fischer theorem,

'jtA) < "jtG22> 5 h,+jtA) (j = 1,2,...,n-r) (1.14)

when

Furthermore, if the columns of the matrix C span the same space as the r

eigenvectors-associated with r smallest eigenvalues of A,

Thus, we see that there is a strong relationship between the eigenvalues

of A and the stationary values of the function

q(x) =xTAx T TT- Ax x + 2p c x,m CI N NN w

where p is a vector of Lagrange multipliers.

(1.16)

l-4

2. Stationary values of a bilinear form subject to linear constraints

NOW let us consider the problem of determining the non-negative stationary

values of

( XTAY >A ll$I,- CI II& 1 (24

where A is an m x n matrix, subject to the constraints

cTx = 0, DTy = 0.N CI (24

The non-negative stationary values of (2.1) are the singular values of A

( i.e., o(A) = [h(~~A)]l/~). It is easy to verify that the non-negative

stationary values of (2.1) subject to (2.2) are the singular values of

where

'CAPD(2.3)

--_

PC = I - cc , PD = I - DD .

The singular values of PCAPD can be computed using the algorithm given in

[93*

Again it is not necessary to compute the matrices PC and PD explicitly.

Lf, as in (l.ll),

C=QTC

D= T&D

2-l

then

T 0 0I;, = % 0 In-$'[ 1 % 5 gJ,a,

where r is the rank of C and s is the rank of D . Then

“(‘C ApD) =CYQEJ&AQEJD&D)

= “(Jc%AGJD) l

Hence if-=_

G = &c AQ;GilG21

G12G22

where GU is rxs and G22is (m-r) x (n-s) 9 then

Thus the desired stationary values are the singular values of G22 .

2-2

38 Some inverse eigenvalue problems

Suppose we are given a symmetric matrix A with eigenvalues

IA 3i !f=l (‘i < ‘i+l> n-land we are given a set of values (A 3-.1 i=l(xi < xi+l) with

Ai < xi < Ai+l . (34

We wish to determine the linear constraint Tc x = 0 so that them-

Tstationary values of x Ax T Tsubject to x x = 1 and c x = 0 T(c c = 1)M N GIN HCI -mare equal to the set of (@~~. From equation (1.5) we have

x= -~(A-h1)-~ c ,m c1

and hence --

Tc x = -~lc~(A-hI)-~ c = 0 . (3.2)NN U

Assuming P f 0, and given A=QAQT where A is the diagonal

matrix of eigenvalues of A and Q is the matrix of orthonormalized

eigenvectors, substitution into (3.2) gives

n

zi=l

with

d2.= 0

1

(3.3)

where Qd = c . Setting A = x (j = 1,2,...,n-1) then leads to am CI j

system of linear equations defining the d2i . We shall, however, give

an explicit solution to this system.

3-l

Let the characteristic polynomial be

n-lPC') = 'n (‘j -'I

j=l

and let

q(A) = fi (Ajj=l

(3.4)

(3.5)

We wish to compute d (dTd = 1) so that q(A) 5 q(A) . ThenN NCI

let us equate the two polynomials at n points. Now

--_rp @k) = .e (xj -Ak) ,

j=l

Hence cp($) = Jr(Ak) for k = 1,2,...,n , if

2dk =

= $ ‘E(Aj-\) .j=lj#k

n-l'n ('j - 'k)j=l . (3.6)

The condition (3.1) guarantees that the right-hand side of (3.6) will be

positive. Note that we may assign dk a positive or negative value so

that there are 2n different solutions. Once the vector d has been

computed, it is an easy matter to compute c.

3-2

We have seen in section 1 that the stationary values of (1.16) inter-

lace the eigenvalues of A. In certain statistical applications [ 4 ] the

following problem arises. Given a matrix A and a set of constraintsTC x = 2,N

we wish to find an orthogonal matrix H so that the stationary values of

(3*7)

are equal to the (n-r)

As was pointed out

values of (3.7) will be

viding the columns of HC

eigenvalues of A . For--_

we see that we may write

C

largest eigenvalues of A .

in the last paragraph of section 1, the stationary

equal to the (n-r) largest eigenvalues of A pro-

span the space

simplicity, we

associated with the r smallest

assume that rank (C) = p . From (l.ll),

Let us assume that the columns of some matrix V span the same space as

eigenvectors associated with the p smallest eigenvalues. We can construct

the decomposition

v =SwT[ 10 ’e where WTW = In and S is upper triangular. Then the constraints

are equivalent to

[RTi O]$x = 0N N

and thus if H is chosen to be

H= Q?

3-3

the stationary values of (3.7) will be equal to the (n-p) largest

eigenvalues of A.

3-4

4. Intersection of spaces

Suppose we are given two symmetric nxn matrices A and B with

B positive definite and we wish to compute the eigensystem for

Ax = ABx .#u w

One ordinarily avoids computing C L-B-1A since the matrix C is not

symmetric. Since B is positive definite, it is easy to compute a

matrix F such that

TF B F = I

and we can verify from the determinantal equation that

A(FTAF) = A(flA) .

(44

The matrixT=.F AF is obviously symmetric and hence one of the standard

algorithms may be used for computing its eigenvalues.

Now let us consider the following example. Suppose

A = [ : , , ] , B=[;=]

where s is a small positive value. Note B is no longer positive

Tdefinite. When x = [l,O,O] , then Ax = Bx and hence A = 1 . WhenNeXT = [O,l,O] , then Ax = E-$X .

- -1Here A = s and hence as s gets

hl

arbitrarily small, A(E) becomes arbitrarily large. This eigenvalue is

I unstable; such problems have been carefully studied by Fix and Heiberger [ 51.

Finally for T = ~o,wl ,x Ax = ABx for all values of A . Thus weCI Nhave the situation of continuous eigenvalues. We shall now examine ways

of eliminating the problem of continuous eigenvalues.

4-l

The eigenvalue problem Ax = ABx can have continuous eigenvaluesCI m

if the null space associated with A and the null space associated

with B intersect. Therefore we wish to determine a basis for the

intersection of these two null spaces. Let us assume we have determined

X and Y so that. .

Ax =O, BY=0

with

Let

XTX=I and YTY = IP Gl. l

z=[x:Y] ..

Suppose H is an nxv basis for the null space of 2 with

Hz .f.--_ [ 1Fwhere E is pxv and F is qxv . Then

ZK =XE+YF=O .

Hence the nullity of Z determines the rank of the basis for the

intersection of the two spaces.

Consider the matrix

L=ZTZ .

Note nullity(L) = nullity(Z) . From (4.3, we see that

L =

I XT1P

YTX Icl

r1T

0q 1

(44

(4.3)

*P+q+ w l

VW

4-2

Since A(L) = A(I+W) = l+ A(W) ,

A(L) = l+ o(T) . (4.5)

Therefore if oj(T) = 1 for j = 1,2,...,t , from (4.5) we see that the

nullity(L) = t . Thus if we have the singular value decomposition

T =XTY =UCd

where

u = [~lY.*~Y~pl Y

tthe vectors (XEROX=, yield a basis for the intersection of the two

spaces. We can use the set of vectors (Xx$tzl to deflate A and B

simultaneously by an orthogonal similarity transformation.--_

The singular values of XTY can be thought of as the cosines between

the spaces generated by X and Y . An analysis of the numerical methods

for computing angles between linear subspaces is given in [2]. There

are other techniques for computing a basis for the intersection of the

subspaces, but the advantage of this method is that it also gives a way

of finding vectors which are almost in the intersection of the subspaces.

4-3

.

5* Eigenvalues of a matrix modified by a rank one matrix

It is sometimes desirable to determine sane eigenvalues of a

diagonal matrix which is modified by a matrix of rank one. In this

section, we give an algorithm for determining in O(n2) numerical

operations sOme or all of the eigenvalues and eigenvectors of

TD+ouu where D = diag(di) is a diagonal matrix of order n .au-

Let C T=D+ouu . . .,MH; we denote the eigenvalues of C by Al,A2, An

and we assume 'i ,< 'i+l and di < di+l . It can be shown (cf. [lb])

that

(1)

(2)

if U>OY di < hi < di+l ( i = 1,2,...,n-1) ,

Tdnk2 k

= ' (di-A) + ~ ' uii=l i=l

' (dj-A)>j=lJ&i

then it is easy to verify that

%+l(A) = (dk+l -A) (p,b) + 0

$k(h) = @k-h) $+1(h) (k = 1,2,...,n-1) (5.2). .

with JlO(h) = cam = 1.

Thus it is a simple matter to evaluate the characteristic equation for any

value of A. Several well-known methods may be used for computing the eigen-

values of C. For instance, it is a simple matter to differentiate the

expressions(5.2) with respect to A and hence determine c&(A) for any

value of A. Thus Newton's method can be used in an effective manner for

computing the eigenvalues.--_

An alternative method has been given in [1] and we shall describe that

technique. Let K be a bi-diagonal matrix of the form

K =

m

1

L

\and let M = diag(+l .

rl

1 r2

.

0

0

. rn-1

1

Then IWE? is a symmetric tri-diagonal matrix

with elements {p rk k-l' ( Pk+Pk+l kr2> +&k&l ("0 = rn = CL,1 -

-0) .

5-2

Consider the matrix equation

T(D+ouu)x=hx . (5.3)

Nh) - u

Multiplying (5.3.) on the left by K and letting x = K?y , we haveN H

K(D+ouuT)$y = AKI;Ty-LI d

or

(KDKTT T

+aKuu K)y= AKI?y .N.-d CI cv

Let us assume that we have reordered the elements of u (and hence of D ,CI

also) so that

=u =...=u = .*a l-=. u1 0

Peters and Wilkinson [ 13 ] have shown how linear interpolation may

be used effectively for cmputing the eigenvalues of such matrices when

the eigenvalues are isolated. The algorithm makes use of det(A- AB)

which is quite simple to ccxnpute when A and B are tri-diagonal. Once

the eigenvalues have been computed it is easy to compute the eigenvectors. .

by inverse iteration. Even if several of the eigenvalues are equal, it is

often possible to compute accurate eigenvectors. This can be accomplished

by choosing the initial vector in the inverse iteration process to be

orthogonal to all the previously computed eigenvectors and by forcing the

computed vector after-the inverse iteration to be orthogonal to the

previously computed eigenvectors. In some unusual situations, however,

this procedure may fail.-e_

The device of changing modified eigensystems to tri-diagaonl

matrices and then using linear interpolation for finding the roots can

be extended to matrices of the form

Again we choose K SO that Ku satisfies (5.4) and thus obtain theCI)

eigenvalue problem Ay = hBy wheree

KKT 0

A = Y B = - -1oT 1m-so that A and B are both tri-diagonal and B is positive definite.

Bounds for the eigenvalues of C can easily be established by the

terms of the eigenvalues of D and hence the linear interpolation

algorithm may be used for determining the eigenvalues of C .

5-4

6. Least squares problems

In this section we shall show how eigenvalue problems arise in linear

least squares problems. The first problem we shall consider is that of

performing a fit when there is error in the observations and in the data.

The approach we take here is a generalization of the one in [g ]. Let A

be a given m x n matrix and let b be a given vector with m components.

We wish to construct a vector f which satisfies the constraintsu

(A + E)x = b + 6

and for which

w-1)

\\P[Ei_G]qI = min (6.2)

where P is a diagonal matrix with pi > 0, Q is a diagonal matrix--.

with qj > 0, and \\...\I indicates the Euclidean norm of the matrix. We

rewrite (6.1) as

or equivalently as

where

By + Fy = 0CI

B = [Aib]Q,

F = [Eia]a,Xz = Q-l -;tI I

(6.3)

(6-4)

Our problem now is to determine y so that (6.3) is satisfied, and

II IIPF =minAgain we use Lagrange multipliers as a device for minimizing (IPFII

subject to (6.3).

6-1

Consider the function

m n+l n+lcp(F) = C C ~;f2~ - 2 ;Ai

i=l j=l i=l' Cbijj=l

+ fij)Y..J

(6.5)

Then

FPf = 2PEfrs - 2A,y, ..rsso that we have a stationary point of (6.5) when

Note that the matrix F must be of rank one. Substituting (6.6) into (6.3)

we have

A =

and hence =.

PF=

Thus,

p2BY“--

(YTY>NN

PB YYTHCI

sr’s:

.

VT9

II IIPF2y&B&P-By

F -(YTY)NN

and hence PFII II = min when y is the eigenvector associated with the smallestNeigenvalue of BTP2B. Of course a more accurate procedure is to compute the

smallest singular value of PE3.

Then, in order to compute 2, we perform the following calculations:M

(a) Form the singular value decomposition of PB, viz.,

PB= u c VT,

(It is generally not necessary to compute U.)

(b) Let v be the column vector of V associated with omin(PB)w

so that v = G . ComputeCI

Z = Qv .N

6-2

(c) From (WY

Note that mini ~~11 =u,,,(P@ , a%-that

L-1-1 1-Z'n+l y[E I s] = - [A :b]vvTg-' .CI N.-d

The solution will not be unique if the smallest singular value is

multiple. Furthermore, it will not be possible to compute the solution

if zn+l

= o . This will occur, for example, if P = Im , Q = Iti ,

ATb = oCI CI

md umin(A) < II t 112 l

Another problem which arises frequently is that of finding a least--._

squares solution with a quadratic constraint; we have considered this

problem previously in [l]. We seek a vector x such thatCI

\\b -A"\\, = min

with the constraint that

II II2c2=a .

The condition (6.8) is frequently imposed when the matrix A is

. ill-conditioned. Now let

q(x) = (b -Ax)T(b -Ax) + tiT x& w N - d

. where A is a Lagrange multiplier. Differentiating (6.9), we are led

to the equation

ATAx - ATb + hx = 0N M w

or

(ATA + AI)x = ATb .cs) N

(6.7)

(6-8)

(6.9)

(6.10)

(6.11)

Note that (6.10) represents the usual normal equations that arise in theT

linear least squares problem, with the diagonal elements of A A

shifted by h . The parameter X will be positive when

and we assume that this condition is satisfied.

Since x = (A~A+~I)-~A~~ , we have from (6.8) that

bTA(ATA+ AI)-* ATb - a* =o .

By repeated use of the identity

det * 'c 3

z w= det(X) det(W 4X%) if det(X) { 0 ,

we can show that (6.12) is equivalent to the equation

det((ATA+hI)* - a-*ATb bTA) = 0 .- hl

Finally if A = Ux? , the singular value decomposition of A y then

ATA=VDVT , &=I

where D =Tc c and (6.13) becomes

det((D+AI)* - uuT) = 0Hh)

(6J-a

(6.13)

(6.14)

(6.15)

where u = a-1 T TC U b . Equation (6.15) has 2n roots; it can be shown

(cf. 161) that we iced the largest real root of (6.15) which we denote

- by h" . By a simple argument, it can be shown that h" is the unique

root in the interval [O,uTu] . Thus wehl OI

an eigenvalue of a diagonal matrix which

As in Section 5, we can determine a

(5.4) and hence (6.15) is equivalent to

have the problem of determining

is modified by a matrix of rank one.

matrix K so that Ku satisfiesCI

det(K(D+XI)*$ - KuuTKT) = 0 .NN(6.16)

6-4

The matrix G(h) = K(D+hI)*? - KuuTKT is tri-diagonal so that it is&N

easy to evaluate G(h) and det G(h) . Since we have an upper and lower

bound on h* , it is possible to use linear interpolation to find h* ,

even though G(h) is quadratic in h . Numerical experiments have

indicated it is best to compute G(L) = K(D+hI)*KT -KuuTKT for eachNN

approximate value of h* rather than computing

G(h) = (KD21;c-KuuTKT)+ 2hKDl?+h*K8 .4v-

Another approach to solve for L* is the following: we substitute

the decomposition (6.14) into (6.12) and are led to the equation

n U2l(p,(A) 5 c

i=l (di+iA)2- 1 = 0 ,

-=.

(6.17)

with u = a-1 T TCub. It is easy to verify that ifN

k k U2

gk(') = 17 (di+k)* Ci

i=l (di+h)*- 11 Yj=l

Jtk+l(‘) = (%+l+‘)* gk(‘) -“;+l$$h) (k=o,l,*mD,n-l) @*18)

$@) = ($+ ‘>* $&) (k=l,*,...,n-1)

with

Jr@ = IO(h) = 1 .

Thus, using (6.18) we can easily evaluate gn(h) and *n(h) , and hence

use one of the standard root finding techniques for determining h* .

It is easy to verify that x = V(D+h*I)-1

CUT b .w CI .

6-5

II.

A similar problem arises when it is required to make

II IIz 2 = min

when

IIF - 912 = B . .where

p > min lib -AX\\ .x - wCI

Again the Lagrange multiplier h satisfies a quadratic equation which is

similar to the equation given by (6.14).

.

6-6

70 Gauss-type quadrature rules with preassigned nodes

In many applications it is desirable to generate Gauss type quadrature

rules with preassigned no&s. This is particularly true for numerical

methods which depend on the theory of moments for determining bounds

(cf. [ 3 3, And for solving boundary value problems [12"]. We shall

show that it is possible to generate these quadrature rules as a modified

eigenvalue problem.

Let w(x) ,> 0 be a fixed weight function defined on the interval

[a,b]. For o(x) it is possible to &fine a sequence of polynomials

P,(X), P,(x),.*. which are orthonormal with respect to w(x) and in

which p,(x) is of exact degree n so that

--.s" P,(X) p,(x) W(x) = 1 when m = n,a

= 0 when mfn.

The polynomial p,(x) = kn n (x-ti), kn > 0, has n distinct real rootsi=l

a < tl < t2 < . . . < tn < b. The roots of the orthogonal polynomials play

an important role in Gauss type quadrature.

Theorem: Let f(x) E C2N[a,b]; then it is possible to determine

positive w. so thatJ

s" f(x) w(x)dx = "c w.f(tj) + R[f]a j=l '

where

[ n (x-ti)]* O(x)dx, a < r\ < b.

Thus, the Gauss type quadrature rule is exact for all polynomials of degree

< 2N-1.

Any set of orthonormal polynomials satisfies a three term recurrence

relationship:

7-l

pjpj(X) = (X-aj)pj_l(x) - ~j-~pj-*'x) for j = 1y2y'*'yN'

p-,(x) f 0, PO(X) = l*

We may identify (7.1) with the matrix equation

xp(x) =&JNP(x) + @$?N(~)"N a-

where

[p_(41T = ~Po(x),pl(x)‘~*.‘~~o~(x)]’

:; = [O,O )..., 11,

and

--.

JN =

5 %

$2

0

0.

. . l 'N-1. .

‘%l TV

Suppose that the eigenvalues of JN are computed so that

JN%j j-j= A q (j = 1,2,...,N)

e with~~~j = '

and

Then it is shown in [ll] that

(7.1)

(7.2)

7-2

t . = A3 j'

wj = (Slj)*'1

(‘79 3)

A very effective way to compute the eigenvalues of JN and the first

component of

Francis (cf.

Now let

that

the orthonormalized eigenvectors is to use the Q,R method of

C141).

us consider the problem of determining the quadrature rule so

J" f(x)ti(x)dx a zM

awjf(tj) + c Vkf(Zk)

j=l k=l

where the nodes (z }M

k k=lare prescribed. It is possible to determine

cw jYtjINY (vkI~=l so that we have for the remainder:-=_

R[fl = w s", k!l(x-vk)[j$x-tj)]2u(x)dx, a < r\ < b.= =

For M = 1 and z1 = a

andfor M=2 with z1

formula.

or z, = b, we have the Gauss-Radau type formula,L

= a and z2 = b, we have the Gauss-Lobatto type

First we shall show how the Gauss-Radau type rule may be camputed. For

convenience, we assume that z1 = a. Now we wish to determine the polynomiale

'N+l x( 1so that

pN+l(a) = ‘*

From (7.1) we see that this implies that

0 = PN+l(a) = (a-gN+l)PN(a) - @NpN-l(a)

or

'N-1 a( >

?N+l=a- N p,(s) l (74

From equation (7.2) we have

7-3

(JN -al)P(a) = - BNpN(a)~~c1

or equivalently,

(J,-aI)h(a)M = /3: eN (7.5)

where . .

Thus,

?N+1 = a+&N(a) . (7 96)

Hence, in order to compute the Gauss-Radau type rule, we do the following:

(a) Generate the matrix JN+l l

(b) Solve the system of equations (7.5) for 6N(a) .

(4 compute y(+1 by (7.6) and use it to replace the (N+l,N+l)element of JN+l .

(a> Use the Q,R algorithm to compute the eigenvalues and first

eigenvector of matrix

Of course, one of the eigenvalues of the matrixJN+l must be equal to a .

Since a < hmin(JN) , the matrix JN- a1 will be positive definite

and hence Gaussian elimination without pivoting may be used to solve (7.5).

- It is not even necessary to solve the complete system since it is only

necessary to compute the element$ca> l

However, one may wish to use

iterative refinement to compute $&') very precisely since for N large,

Amin (J) may be close to a and hence the system of equations (7.5) may

be quite ill-conditioned.

7-4

When z1 = b, the calculation of iN+l is identical except with b replacing

a in equations (7.5) and (7.6). The matrix JN - b1 will be negative

definite since b > Amax(

To compute the Gauss-Iobatto quadrature rule, we need to compute a. .

matrix 'N+lsuch that

N

Thus, we wish to determine pN+l(x) so that

'N+l a( 1 = PN++) = 0.

Now from (7.1) we have

---'N+l'N+l(x) = (x - ~+l)pN(x) -

so that (7.7) implies that

a~+lPN(a) + f$$?N-1(“> = ‘+$ca>

s+lpNtb) + f$$?N-l(b> = b??ly(b)

Using the relationship (7.2), if

(JN- -aI)X = zN 1and

(JN- dbI)p = zN

. then

(7.7)

BNpN-l x ’( 1

(j = 1,2,...,N).

( 79)

(7.10)

Thus, (7.8) is equivalent to the system of equations

7-5

TV+l - ANpi = a

%+l- pN$=b.

( 7.11)

Hence, in order to compute the Gauss-Lobatto type rule, we perform the

following calculations: . .

(a) Generate the matrix JN l

(b) Solve the systems of equations (7.9) for AN and pN.

(c) Solve ( 7.11) for a~+~ and &.

(d) Use the QR algorithm to compute the eigenvalues and first element

of the eigenvectors of the tridiagonal matrix

'N+l =JN 'N:N[t-l.-e_ 'N$f ?N+l

Galant [7 ] has given an algorithm for computing the Gaussian type

quadrature rules with preassigned nodes which is based on a theorem of

Christoffel which gives a method for constructing the orthogonal polynomials

with respect to a modified weight function.

7-6

ACKNOWLEDGMEXTS

The author wishes to thank Mr. Michael Saunders for making several

helpful suggestions for improving the presentation, Mrs. Erin Brent for

performing so carefully some of the calculations indicated in Sections 5

and 6, and Miss Linda Kaufman for performing numerical experiments

associated with Section 7.

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Dl

RI

c31

P+l

151

161e

c71

WI

[91

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Ke?r Words

EigenvaluesNumerical linear algebraSingular valuesLeast squaresInverse eigenvalue problemsQuadrature rulesStatistical calculations